89

CHAPTER 6

MODELING OF ABSORPTION OF RADIANT

ENERGY

6.1. BACKGROUND

The most dominant obstacle in understanding the exhaust plume is that the

radiative properties, namely, the extinction and absorption coefficients, the scattering

coefficient, the temperature distribution and the scattering phase function of the plume are

generally unknown. The modeling of reduction of intensity of radiant energy while it travels

through a medium due to the absorption is the main theme of this chapter. Scattering is

often accompanied by absorption. Both scattering and absorption remove energy from the

radiant energy traversing the medium; then the energy is said to be attenuated. This

attenuation is called extinction. In this chapter, the process of absorption of radiant energy

by the mixture of gases present in the combustion products of the exhaust plume is

explained. The absorption characteristics of both liquid and solid phase alumina are

discussed, as alumina is one of the major constituents of the combustion products of solid

rockets.

6.2. INTRODUCTION

In the domain of exhaust plume of solid rockets, the combustion gases are the main

component which absorbs the radiant energy. If the radiation properties of gases and

opaque solids are compared, the property variations with wavelength for opaque solids are

continuous whereas gas properties exhibit very irregular wavelength dependences. The

absorption or emission by gases is significant only in certain wavelength regions.

Differences in emission spectra are caused by the type of energy transition that occur with a

particular medium. Since the absorption coefficient is a function of density, gases posses a

highly variable absorption coefficient.

90

Many problems of radiative heat transfer involve the exchange of radiant energy

between surfaces separated by a medium that absorbs and emits radiant energy. Most

participating media may have radiative properties that vary pronouncedly with the

frequency characterizing the radiation. One of the important issues for radiative heat

transfer in a combustion system is the description of the radiative properties of particle laid

combustion gases. Beer’s law was originally established as a result of experimental studies

of the attenuation of light in colloidal suspensions. Early studies in this area were made by

Bouguer in 1729 and later by Lambert. The law states that the intensity of radiant energy

traversing a semitransparent medium is decreased by absorption in proportion to the

intensity at that point. In thermodynamic equilibrium, the intensity of radiation does not

change along the path. The website http://Wikipedia reveals the contribution of both Beer

and Lambert as follows :

Lambert law states that absorption is proportional to the length of the path,

whereas, the Beer law states that absorption is proportional to concentration of absorbing

species in the material. If the concentration is expressed as a mole fraction, the molar

absorptivity takes the same as the absorption coefficient, m-1 .

From quantum theory, it is known that that the structure of a single molecule is such

that it may emit and absorb only certain energy quanta corresponding to the allowable

energy configurations of the molecule. Thus a gas composed of identical molecules would

absorb and emit photons having exact frequencies corresponding to these allowable

energies. A radiating gas like the combustion products of a solid motor is composed of

molecules, atoms, ions and free electrons which are associated with different energy levels.

It is convenient to discuss the radiation process by utilizing a photon, which is the basic unit

of radiative energy. Radiative absorption is the capture of photons while emission is the

release of photons.

Most solids and liquids absorb energy in rather broad wave length range. Gases on

the other hand, tend to absorb and emit energy in narrow bands that are only a few microns

in width, called a line. The problem of integrating over frequency must therefore be treated

in a different fashion than in the surface radiation problem. In real gases, these emission

lines must have finite width and are shaped as a probability function. The combination of

several phenomena results into the line broadening effect. The phenomena leads to the line

broadening effect are the following:

91

(1) Natural line broadening: This is resulted from the variation of emission levels from

several photons from a large number of molecules.

(2) Collision broadening: This is resulted from the interaction of the force fields between

molecules. Transition energies are affected by the proximity of colliding molecules

and thus the frequency of the photon emitted or absorbed in the transition is

influenced. The collision frequency is influenced by the pressure and temperature of

the gas. As the gas pressure is increased, the emission lines broaden into bands.

(3) Doppler broadening: It is the result of movement of the molecules or atoms. Since

the gas is in a state of thermal agitation, photons will be emitted over a small range of

frequencies as a result of this effect.

(4) Stark broadening: this occurs as a result of the interacting electrical fields of ions and

the free electrons present in high temperature gases or plasmas.

Recently, Guo-Biaco Cai etal [2006] proposed a weighted sum of gray gases

model, which replaces the nongray gas behavior by equivalent finite number of gray gases.

The above discussions reveal the complexities of modeling absorption in a mixture

of gases. Thus in most of the engineering applications, the spectral effects in absorption

phenomena of gases are not taken into account. This is justified to some extent because of

the prevailing lower absorption coefficients at higher temperature levels of radiating gases.

By assuming the condition of optical thin limit and gray gas properties, an appropriate

averaging procedure for the mean absorption coefficient of the mixture of gases is used in

this analysis.

6.3. ABSORPTION

The attenuation of radiant energy inside a gas of infinitesimally thin thickness ds is

proportional to the radiation intensity I and thickness ds and can be expressed as

dI ak I ds= − (6.1)

Here the constant ak is called the gas absorption coefficient. This equation is

valid, exactly, for only a monochromatic ray whose absorptivity is a function of the

wavelength, temperature and pressure. In the case of radiant energy of a broad wavelength

band, the above equation is valid only approximately. Equation (1) is linear and the solution

is

92

0

akSI I e= − (6. 2)

Where I0 is the initial intensity and I is the resultant intensity after passing through a

distance of S meters in an absorbing medium with absorption coefficient ak . Here the

attenuation factor of the intensity is akSe−

and the product ak S is known as the

absorptive distance or optical length, which is dimensionless.

The absorbing properties of the gases can change by several orders of magnitude

over a small wavelength interval. Another difficulty also arises from the nature of the

monochromatic absorption coefficient being non-continuum and possessing a strong

functional dependence on the total pressure. An appropriate averaging of absorbing

properties over the wavelength spectrum is required to arrive at meaningful radiant heating

rates. Abu_Romia and Tien [1967] have proposed the planck mean absorption coefficient

for gases that approach the optical thin limit. Gas is optically thick in some parts of the

infrared spectrum and optically thin or almost transparent in the rest.

6.4. PLANCK MEAN ABSORPTION COEFFICIENT

Planck means absorption coefficient is a widely accepted method for estimating the

absorption coefficient of gases. The line intensity spacing ratio is a property of absorbing

gases. Therefore in a mixture of absorbing and inert gases, it could be concluded that the

Planck mean absorption coefficient per unit optical depth is independent of the inert gas

pressure and only varies with temperature. Thus Planck means absorption defined by eqn.

(6.3) is used in the analysis

,

0

,

0

, ( , , ) ( , )

( , )

( , )

a b

b

K T P E T d

Ka T P

E T d

λ λ

λ

λ λ λ

λ λ

∞

∞=∫

∫ (6.3)

Here, the spectral variation of the volumetric absorption coefficient is weighted by

the blackbody function. Thus the plank mean absorption coefficient will be

(1) Independent of wavelength

(2) A function of the partial pressure

93

(3) A function of the temperature

Figures 6.1, 6.2 and 6.3 give the plank mean absorption coefficients of CO, CO2 and H2O

respectively, calculated for a Partial pressure of 1atm.

0

1x10-5

2x10-5

3x10-5

4x10-5

500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000

Planck mean absorption coefficient of CO

TEMPERATURE, K

PL

AN

CK

ME

AN

AB

SO

RP

TIO

N C

OE

FF

ICIE

NT

(Pa.m

)-1

Figure. 6.1 Planck mean absorption coefficient of CO

0

0.0001

0.0002

0.0003

0.0004

0.0005

500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000

Planck mean absorption coefficient of CO2

TEMPERATURE, K

PL

AN

CK

ME

AN

AB

SO

RP

TIO

N C

OE

FF

ICIE

NT

(Pa

.m)

-1

Figure. 6.2 Planck mean absorption coefficient of CO2

94

0

0.00005

0.00010

0.00015

0.00020

500 1000 1500 2000 2500 3000

Planck mean absorption coefficient of H2O

TEMPERATURE, K

PL

AN

CK

ME

AN

AB

SO

RP

TIO

N C

OE

FF

ICIE

NT

(P

a.

m)-1

Figure. 6.3 Planck mean absorption coefficient of H2O

For calculation in thermal radiation, it is necessary to ratio these coefficients from

1atm to the partial pressure of the gas being modeled. For a mixture of gases, the Planck

mean absorption coefficient, ak, is defined as

Ka, mixture = ( P/1atm) ∑=

4

1

.i

ii FMKa (6.4)

Where Kai is the Linear absorption coefficient of the radiating gaseous components like

CO,CO2,HCL and H2O and M.Fi is the Mole fraction. The mole fractions of the principal gas

emitters are influenced by the method with which the plume chemistry is modeled. Values

are provided by using equilibrium chemistry. The predicted values of mole fractions of

combustion products at the exit plane of a typical solid rocket are given below:

Table 6.1: Mole fractions of combustion products

Name of combustion product Mole fraction

H2 0.310 Al2O2 Gas species 0.227

HCl 0.155

H2O 0.119

Al2O3 solid phase 0.089

N2 0.079

H 0.003

CO2 0.016

Cl 0.001

95

The Planck mean absorption for the above mixture of combustion gases computed

by the eqn (6.4) is shown in fig. 6.4

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

Temperature, K

Ab

so

rpti

on

co

eff

icie

nt,

m-1

Figure. 6.4 : Planck mean absorption coefficient

for mixture of gases given in table 6.1 It may be noted that the Planck mean absorption coefficient of HCl is taken to be

that of CO, due to the unavailability of data. This approximation is based on the following

reasons as reported by Watson and Lee (1976).

(1) Spectral absorption coefficient for HCl is lower than that for CO.

(2) The band for HCl subtends a greater fraction at the blackbody function than

CO.

These two effects may mitigate against each other so that the above approximation

is reasonably valid.

6.5. ABSORPTION CHARACTERISTICS OF Al2O3

In the molten state, Carlson DJ (1967), compiled data for the non-dimensional

absorptive index is represented by the equation

)(

,0T

TT

maa

m

ekk∆

−

= (6.5)

where ka,m=2*10-4 and ∆T0 =260 K and Tm=2320 K.

96

Carlson’s review indicates a two order of magnitude drop in ka upon freezing. Thus

a solid particle at the very same elevated temperature (2320 K) as a fully molten particle

has negligible emission in comparison. It may be noted that the absorptive characteristics of

molten alumina up to 2500 K is negligible and hence the absorption of both molten alumina

and solid particles upon freezing is not taken into account in the analysis of plume radiosity.

The absorption efficiency for a size parameter (ψ=2πr/λ) in the range 4<=x<=20 and for a

refractive index 1.78, result gives Qa = 11.7 ka *0.68. Both Ka and Qa are plotted.

Figure 6.6 shows the absorptive characteristics of molten aluminia to show the

difference of absorption characteristics of alumina with phase change. Since, the presence

of molten alumina is not reported in the exhaust plume of bigger motors, its effect is not

taken in the analysis.

-0.00001

0.00004

0.00009

0.00014

00.2x10

-50.4x10

-50.6x10

-50.8x10

-51.0x10

-5

1000 0C

100 oC

Wave Length, λλλλ (m.)

Imag

inary

Part

of

Refr

acti

ve

In

dex

Figure. 6.5 Variation of absorption coefficients of solid Al2O3 particles at elevated temperatures.

0

0.05

0.10

0.15

1750 2000 2250 2500 2750 3000 3250 3500

Absorption EfficiencyAbsorptive Index

Temperature, K

Absorp

tive C

hare

cte

ristics

Figure. 6.6 Absorbptive characteristics of Molten Alumina

97

6.6. CONCLUSION

Absorption of solid particles is neglected. Planck mean absorption coefficient presented in figure 6.4 will be used as the absorption coefficient of the mixture of gases in the exhaust plume.

98

CHAPTER 7

MODELING OF SCATTERING OF RADIANT

ENERGY

7.1. BACKGROUND

Modeling of thermal radiation in participating media necessitates the modeling of

scattering of radiant energy due to solid particles to account the redistribution of radiant

energy in the domain of the system. Scattering is the parameter which influences mostly the

quantum of radiant energy which escapes the exhaust plume boundary of the rockets and a

part of this energy is incident on the base region of the rocket depending on the view factor

of the region of escape from the boundary to the base region of the rocket. Thus modeling

of scattering of radiant energy emanating from each control volume of the computational

domain of the exhaust plume became essential.

The scattered energy depends on wavelength of incident radiant energy, the size

and shape of the scatterer and refractive indices of the scatterer and the medium in which

the scatterer is located. These parameters are interconnected through complex

mathematical functions, infinite series and recurrence relations. Hence designers usually

resort to make approximations for the particle size parameter and wave length to estimate

the scattering coefficients. Such an approximation becomes erroneous since the scattering

coefficients are highly non-linear and sensitive to the size parameter as will be shown in this

chapter. Further, the scattering efficiency predicted by the approximate formulae cannot

exhibit all minor wiggles and some times these formulae are not available in the required

area of interest. Hence a rigorous approach is required to capture all the wiggles in the

curve of scattering efficiency factors and for larger size parameters for achieving better

accuracy for the results.

Phenomenon of scattering and the built in natural size parameter of the scattering is

explained. Influence of parameters such as refractive indices of scatterer and medium, size

parameter and absorption and scattering characteristics of particle is described. Scattering

99

diagrams and phase functions for diffuse reflecting spheres are introduced as applicable

cases in the present study for predicting scattering efficiencies for aluminium oxide particles

contained in the exhaust plume of solid rockets. The directional variation of scattered

radiation intensity as predicted by the Rayleigh phase function is presented. The

scattering patterns of small non absorbing spheres and that of a sphere with a

Lambertonian surface are also shown.

The magnitude of scattering as described by Rayleigh-Debye is presented for

predicting scattering efficiency of a particle which is not too large and when the refractive

index of the particle is sufficiently close to that of the external. Finally Mie theory of

scattering is explained which is valid in a larger domain by defining the scattering

coefficients as functions of Ricatti-Bessel functions and their derivatives. The scattering

efficiency factor is predicted using a series expansion of scattering coefficients. To make the

computation more efficient, the Ricatti-Bessel functions and their derivatives are expressed

as polynomial functions of size parameter and this approach realized the convergence of

the value of scattering efficiency with just seven terms in the series expansion of scattering

coefficients. This mathematical model of predicting the scattering efficiency can thus be

used an input for various analysis in other scientific fields as a subroutine which delivers the

input with minimum computational time.

The developed mathematical model is validated in a number of cases using an

approximate formula derived by Van de Hulst [1981], which is valid over a size parameter

range up to 2 and thus could extend to higher range of the size parameter of Mie scattering.

The scattering efficiency of a control volume with particles of different sizes is then defined

for the solution of radiant transport equation in the computational domain.

7.2. INTRODUCTION

Theory of scattering has got wide applications in various fields of science and

technology like space sciences, thermal radiation in participating media, image processing,

atmospheric sciences and remote sensing. Everyone engaged in the study of light or its

industrial applications meets the problem of scattering. Scattering is composed of both

diffraction and reflection. The scattered energy depends on the wave length of incident

energy, the size and shape of the scatterer and refractive indices of scatterer and the

100

medium in which the scatterer is located. These parameters are usually interconnected

through complex mathematical functions, infinite series and recurrence relations. Both the

physics and mathematical models of scattering of radiant energy are complex.

Even though the study of scattering of electromagnetic fields was initiated in the last

century, especially in the field of optics, its application in thermal radiation remains as an

area where much detail are not readily available. Hence the modeling of scattering cross

section in the field of thermal radiation became a challenging task. “Except for conference

tutorials and a few isolated projects and classroom examples, very little is formally taught

about the subject” as opined by John C Stover [1990]. Many of the topics presented in the

discussion of Mie theory is taken from the classic reference of Van de Hulst [1981]. Since

considerable time of this study has been spent for modeling scattering of radiant energy

alone, full details of the study is presented for the benefit of future research work in various

fields including optics.

Scattering is often accompanied by absorption. A leaf of a tree looks green

because it scatters green light more effectively than red light. The red light incident on the

leaf is absorbed. Both scattering and absorption remove energy from a beam of light

traversing the medium. The beam is said to be attenuated. This attenuation is called

extinction and thus extinction is the sum of scattering and absorption.

Both absorption and scattering of light energy is applicable in its full meaning to

thermal radiation also as it is part of the electromagnetic radiation. If electromagnetic energy

travels through a perfectly homogeneous medium, it is not scattered. Only inhomogenity

causes scattering. The presence of particles or small scale density fluctuations, make part

of the radiation scatter in all directions. Any material medium has inhomogenities as it

consists of molecules, each of which acts as a scattering center, but it depends on the

arrangement of these molecules whether the scattering will be very effective. In a gas or

fluid, statistical fluctuations in the arrangement of the molecules cause a real scattering,

which sometimes may be appreciable. In this case, whether or not the molecules are

arranged in a regular way, the final result is a co-operative effect of all molecules. Then one

has to investigate in detail the phase relations between the waves scattered by neighboring

molecules. The precise description of the co-operation between the particles is called

dependent scattering and is not addressed in this study.

101

In the exhaust plume of solid rockets, Liquid particles of Al2O3 possess relatively

large absorption cross sections while solid particles are almost pure scatterers of radiant

energy. However, the presence of liquid particles of Al2O3 in the exhaust plume is almost

negligible and solid particles of varying sizes are present. The solid particles of Al2O3 are

having both absorption and scattering cross sections. Different methods of computations

exist as can be seen in Edwards and Baikiant [1990] and Gilbert N Plass [1966] in

calculating scattering efficiency factors depending on the nature of refractive index of Al2O3.

7.3. COMPLEX REFRACTIVE INDEX OF Al2O3

When the spherical particles are considered as absorbing, the refractive index is

defined by a complex number as m = n1-in2 where 1−=i . The real and imaginary parts

represent the scattering and absorption coefficients respectively. The influence of n1 and n2

on the Mie scattering and absorption coefficients are discussed in detail by Gilbert N. Plass

[1966]. Table-7.1 gives the complex refractive index of Al2O3 as a function of wavelength, λ

Table 7.1 Complex refractive index of Al2O3

Wavelength,

λ(µm) n1 n2

0.5 1.77 1.0E-06

1.0 1.75 1.0E-06

2.0 1.74 1.0E-06

3.0 1.71 1.0E-06

4.0 1.68 1.0E-05

5.0 1.63 1.0E-04

6.0 1.54 2.2E-04

8.0 1.35 3.3E-04

10.0 1.09 5.0E-04

The sensitivity of imaginary part of complex refractive index, namely the absorption

coefficients of Al2O3 on the wave length as a function of temperature is shown in fig.7.1.

Even though there is appreciable increase in the absorption coefficient at elevated

temperatures in the range of 0.25µ to 0.06µ, the absolute value of absorption coefficient is

102

negligible when compared to the scattering coefficients listed in table 7.1. Hence the

refractive index of Al2O3 particles is considered as real in this study.

-0.00001

0.00004

0.00009

0.00014

00.2x10

-50.4x10

-50.6x10

-50.8x10

-51.0x10

-5

1000 0C

100 oC

Wave Length, λλλλ (m.)

Ima

gin

ary

Pa

rt o

f R

efr

ac

tiv

e In

de

x

Fig.7.1 Imaginary part of refractive index of Al2O3 at normal

and elevated temperatures

7.4. SCATTERING ANGLE AND CROSS SECTIONS

The scattering of radiant energy in the exhaust plume is modeled as follows:

(1) Using the phase function, the direction of the scattering is predicted.

(2) Depending on the size parameter, either Rayleigh scattering or Mie scattering

formulations are used to estimate the scattering efficiency of a particle.

(3) The scattering efficiency of particles in a control volume is then estimated by

adding the scattering efficiency of each particle contained in the control volume. The particle

density of each category of particles is given by the Rosin-Rammler distribution.

The scattered wave at any point in the distant field has the character of a spherical

wave in which the energy flows outward from the particle. Scattering angle is defined as the

angle between the forward direction of the incident beam and a straight line connecting the

scattering particle and the detector. A schematic of the scattering angle is shown in fig.7.2

103

Fig.7.2 Schematic of scattering angle caused by a scattering particle

To fix the point P in a three dimensional space one needs another angle Φ, called

azimuth angle as described in the figure 7.3 Draw Perpendicular to the YZ plane from the

point P. The Perpendicular will intersect the YZ plane at Q. The angle made by OQ with OY

is the azimuth angleφ . If r is the distance OP, then the intensity, I, of the scattered light at

P is given by equation (7.1)

0II = F ),( φη (7.1)

K2r2

where I0 is the Intensity at origin O and λ

π2=K , the wave number. The Function

F ),( φη is the dimensionless function of the direction. The relative values of I or F may be

plotted in a polar diagram as a function of η in a fixed plane through the direction of

incidence. This diagram is called a scattering diagram of the particle and this presents a

total view of nature of scattering as a function of η. Let the total energy scattered in all

directions be equal to the energy of the incident wave falling on the area Csca of the particle.

The term Csca is known as scattering cross section of the particle. It is shown [1] that the

scattering by any finite particle is fully characterized by its four amplitude functions which

are complex functions of the directions of incidence and of scattering. Knowledge of these

amplitude functions suffices for computing the intensity and polarization of scattered light,

the total cross sections of the particle for scattering, absorption and extinction.

η

o Point P

Incident Light

104

Fig.7.3 A schematic of geometry in fixing a point in 3 dimensional spaces.

For homogeneous spheres only the first two amplitude functions are to be

evaluated as functions of scattering angle. When the function F ),( φη is divided by K2Csca,

another function of direction known as phase function is obtained. Directional magnitude of

the scattered intensity is related to the entire scattered intensity times the phase function

Robert Siegel, and John R. Howell (2002)

7.5. SCATTERING PHASE FUNCTIONS

Energy absorbed is converted primarily to thermal energy, whereas scattered

radiant energy is redistributed in the medium. The mapping of redistribution of radiant

energy in the medium is given by the scattering phase function. The scattering function

may be determined analytically for simple geometric shapes, such as spheres, by solving

Maxwell’s equations with the coordinate system and boundary conditions corresponding to

an infinite plane wave incident on the object. This finding was first published by Gustav Mie

[6] in connection with his studies of colloidal suspensions of gold. Scattering Coefficient is

the integral of scattering function over the full solid angle 4π.

For axially-symmetrical scattering function, i.e. scattering function that depends only

on the scattering angle, the phase function denoted by λφ with unit of [Steradian

1] is

obtained by normalizing the scattering function by the scattering coefficient, Qsca .

105

ie; λφ (θ) = scaQ

)(θβλ (7.2)

The phase function can be regarded as a distribution of probability of a photon being

scattered at an angle. For an axially symmetrical phase function [7] we have

2 θθθφπ λ dSin∫Π

0

)( = 2 ∫Π

0

)(

scaQdSin θθθβπ λ =

sca

sca

Q

Q = 1 (7.3)

where the factor of 2π arises from integration over the angle in its full range: from 0 to 2π.

Thus the directional distribution of scattered energy is expressed by the phase function. In

general it is denoted as ),( ΩΩ′λφ , which represents the fraction of energy incident in

direction ,Ω′ and scattered into direction Ω.

Here the angle of incidence is said to be Ω /( η /, φ/). Similarly the angle of the scattered

beam also can be described by Ω ( η , φ). The angle between the direction of the incident

Angle between beam Ω / and the scattered beam Ω is called the scattering angle Θ. The

scattering angle is related to the azimuthal and zenith angles via the following, John R.

Howell and M. Pinar Menguc [1988], equation.

)cos(sinsincoscoscos φϕηηηη ′−′+′=Θ (7.4)

Fig.7.4 Parameters of solid angle Ω / defined by the azimuth angle η / and polar angle φ/

106

The scattering phase function is normalized so that the sum of scattered energy in all

directions add up to 100 percent as shown by equation (7.5)

1sin),,,(4

1),(

4

1

0

2

0

4

0

=′′=ΩΩΩ′ ∫∫∫==Ω

ϕηηϕηϕηφπ

φπ

π

λ

π

ϕ

π

λ ddd (7.5)

The phase function is a measure of the anisotropy of the scattering. It provides a factor for

each direction with which the incoming intensity has to be multiplied to give the outgoing

intensity. Hence for isotropic scattering the phase function is 1 for all direction.

It can be seen that with increasing z, the size parameter, the particles become more

forward scattering. Also for non homogeneous spheres, especially where there is a size

distribution, the small variation on the complex index of refraction does not affect the phase

function significantly. The scattering phase function is purely a function of η for a particle,

such as a sphere, where scattering characteristics are independent of the circumferential

direction φ. If the size is small, say 05.0≤z , then the Rayleigh phase function [5] defined

by equation (6.6) can be used.

( )ηηφ 2cos14

3)( += (7.6)

Figure 7.5 given below shows the directional variation of scattered radiation

intensity as predicted by the Rayleigh phase function. Maximum forward scattering is

occurred at 1800 while the maximum backward scattering occurred at 00. The relative

intensity distribution of the scattered radiant energy is expressed by the ratio of the

scattered radiation intensity to the intensity of isotropic scattering in the same direction.

1=φ is the value for isotropic case for all angles and is plotted in the same figure for

comparison.

On the other hand, for particles with diameter much larger than the wave length of

radiation, ie; for large size parameter, a Legendre polynomial expansion may yield better

accuracy, Milton Kerker [1969], which can account for the strong forward scattering peak.

But this leads to a significant increase in computational effort.

107

Fig 7.5 Scattering phase function for Rayleigh and isotropic scattering

If the intensity of the thermal radiation emitted from a surface does not vary with

direction it is referred to as a Lambertonian surface. The scattering pattern of a sphere with

a Lambertonian surface (also called as diffuse sphere) has been computed by E

Schoenberg [1929] the derived phase function is given by equation (7.7)

( )ηηηπ

ηφ cossin3

8)( −= (7.7)

The scattering diagram of the phase function given by the equation (7.7) is shown in fig.7.6.

The validity of this phase function is further discussed in the literature as valid for a gray

sphere with a radius sufficiently greater than the wavelength.

For small non absorbing spheres the phase function is given by equation (7.8) and the

scattering diagram is given in figure 7.7.

−+

−=

22

2

1coscos

2

11

5

3)( βββφ (7.8)

00

0.5

0.5

1.0

1.0

1.5

1.5

00 0.50.5 1.01.0 1.51.5180

o

135o

90o

45o

0o

315o

270o

225o

Isotropic Rayleigh Scattering

108

00

2

2

4

4

6

6

00 22 44 66180

o

135o

90o

45o

0o

315o

270o

225o

Isotropic ScatteringPhase functionfor a diffuse sphere

Fig 7.6 Scattering phase function for a diffuse sphere and isotropic scattering

00

1

1

2

2

3

3

00 11 22 33180

o

135o

90o

45o

0o

315o

270o

225o

Isotropic scattering functionPhase function of small nonabsorbing spheres

Fig 7.7 Scattering phase function for small non absorbing spheres and isotropic

scattering

7.6. MAGNITUDE OF SCATTERING

A built in parameter known as size parameter is defined in the scattering theory

connecting the geometrical size and the wave length of the incident energy. The size

109

parameter is defined as λ

π rz

2= and the magnitude of scattering is modeled in different

forms depending on the value of size parameter. The limits of validity of scattering formula

depend on both the size parameter and the refractive indices of the particle and surrounding

medium. Clearly the size parameter is a scaling parameter relating the particle diameter

and the radiation wave length. For 3.00 ≤≤ z Rayleigh-Debye scattering theory is valid.

7.6.1. Rayleigh Scattering

If the refractive index of the particle is sufficiently close to that of the external

medium and if the particle is not too large, each volume element behaves as a Rayleigh

scatterer. Each of the scattered wavelets in turn mutually interferes. The model was

precisely calculated by Lord Rayleigh as follows. According to this law, the scattering

intensity is proportional to λ-4 and on increasing the size of the particles, the scattering

intensity changes more slowly as a function of wavelength. The scattering cross section,

Csca, of a sphere with a dielectric constant ε2

( )4

112

657

12

32

2

λεε

πεε

+

−=

rCsca (7.9)

where ε1 is the dielectric constant of the medium and λ is the wave length of radiant energy.

If the radius of the sphere is comparable with the wave length, then the quantity Qs

oscillates with the rising λ around a mean value of the same order as the area of the

projection of sphere.

7.6.2. Mie Scattering

Here in this study, rigorous scattering theory for spheres of arbitrary size is carried

out in order to capture all the wiggles exhibited by the scattering cross sections of

aluminium oxide spherical particles contained in the exhaust plume. In the field of scattering

theory, perhaps Mie scattering may be the branch which finds majority of the applications. It

captures all the wiggles in the domain of scattering by making use of the asymptotic

behavior of the spherical Bessel functions. Both the physics and the mathematical models

are complex and hence designers usually resort to make approximations for the size

parameter so that the Mie scattering coefficient is taken for the model value of the size

parameter in the computational domain. The scattering and extinction coefficients are

functions of the size parameter z. The intensity scattered in all directions for an incident

110

wave of unit intensity is independent of the state of polarization of the incident beam and is

known as the scattering cross section Csca. The size parameter is a built in single parameter

connecting the size of the scatterer and the wave length of the incident radiation. The

refractive index of the scatterer and medium and the size parameter are interconnected

through complex mathematical functions, infinite series and recurrence relations in the

computation of scattering cross section. The scattering cross section is given by

∑∞

=

++=1

222

)12(2 n

nnsca banCπ

λ (7.10)

The efficiency factor Qsca is the ratio of scattering cross section to he geometrical cross

section and is given by

∑∞

=

++=1

22

2)12(

2

n

nnsca banz

Q (7.11)

The average geometrical cross section of a convex particle with random orientation

is one-fourth its surface area (PP:110 Van de Hulst).

The total intensity abstracted from the incident beam of unit intensity both by

scattering and absorption is the extinction cross section and is independent of the state of

polarization of the incident beam. The extinction coefficient is given by

∑∞

=

++=1

2

)Re()12(2 n

nnext banCπ

λ (7.12)

The corresponding extinction efficiency factor is obtained by dividing the extinction

coefficient by geometrical cross sectional area

∑∞

=

+=1

2)12(

2

n

ext nz

Q )Re( nn ba + (7.13)

The coefficients an and bn are[1,9] given by

)15.7()()()()(

)()()()(

)14.7()()()()(

)()()()(

zzm

zzmb

zmz

zmza

nnnn

nnnn

n

nnnn

nnnn

n

′−

′

′−

′=

′−

′

′−

′=

ξβψβψξ

ψβψβψψ

ξβψβψξ

ψβψβψψ

111

Here λ

π rz

2= and

2

1

m

mmandmz ==β where r is the radius of particle and λ is the

wavelength in the medium. Here an and bn are known as scattering coefficients. The

functions )()( zandz nn ξψ are expressed in terms of half integral order Neumann and

Hankel functions as given below:

)(2

)(2

1 zJz

zn

n+

=π

ψ (7.16)

where )(2

1 zJn+

is half integral order Bessel Function.

)(2

)()()(2

1)2( zHz

zizz nnnn +=+=π

χψξ (7.17)

where )(2

1)2(zH n+ is the half integral order Hankel function of the second kind. The

function )(2

)(2

1 zNz

zn

n+

−=π

χ (7.18)

Where )(2

1 zNn+

the half integral is order Neumann function and will be used in the

computation of scattering efficiency for spheres with no absorption. Both the functions

)()( zandz nn ξψ are known as Ricatti functions. More precisely the function )(znψ is

known as Ricatti-Bessel function while the function )(znξ is known as Ricatti-Hankel

function. In the case of refractive index with no component of absorption for a sphere, the

scattering efficiency is defined as

)sin)(sin12(2 2

1

2

2 n

n

next nz

Q βα∑∞

=

++= (7.19)

where the angles nn and βα are given by the equations (7.20) and (7.21) given below

)21.7()()()()(

)()()()(tan

)20.7()()()()(

)()()()(tan

zzm

zzm

zmz

zmz

nnnn

nnnn

n

nnnn

nnnn

n

′−

′

′−

′−=

′−

′

′−

′−=

χβψχβψ

ψβψψβψβ

χβψχβψ

ψβψψβψα

112

Here β and βn are different. The parameter β is already defined as mz=β . In the notation,

βn, n represents the order of the term in eq.(7.21). For example, the values of β1, β2, β3 , β4,

etc. are needed in equation (7.19) which can be evaluated from tan β1, tan β2, tan β3, tan

β4,etc.

7.7. BESSEL FUNCTIONS

Bessel’s function is defined by

0)()1()(1)(

2

2

2

2

=−++ zzz

n

dz

zdz

zdz

zzdn

nn (7.22)

Since Bessel’s function is a second order differential equation, it has two independent

solutions. They are known as Bessel function )(zJ n and Neumann function )(zN n each

of order n (Milton Abramowitz and Irene A. Stegun [1972]) and argument z.

The Bessel function is defined as

mn

m

m

n

z

nmmzJ

2

0 2)1(!

)1()(

+∞

=

∑

++Γ

−= (7.23)

Whenever n is not an integer, the Neumann function is defined as

[ ])()cos()()sin(

1)( zJnzJ

nzN nnn −−= π

π (7.24)

The term Bessel function is frequently used as a generic name for both solutions which are

then called Bessel functions of the first and second kind, respectively. Both the solutions in

general are also called as cylinder function and denoted as )(zzn . There exists recurrence

relations valid for all cylinder functions and for their derivatives and are used in the

computation of both Ricatti functions. The following recurrence relations defined by

equations (7.25) and (7.26) are employed.

)(12

)()( 11 zZz

nzZzZ nnn

+=+ +− (7.25)

)(2

1)(

2

1)(11 zZzZ

dz

zdZnn

n

+− −= (7.26)

113

The most efficient form for computation of Ricatti-Bessel functions are given by the

equations (7.27) and (7.28) as given below.

z

z

zdz

dzz

n

n

n

sin)( 1

−= +ψ (7.27)

z

z

zdz

dzz

n

nn

n

cos)1()( 1

−= +χ (7.28)

The first two terms of order 0 and 1 of each Ricatti- Bessel function [9] are given as

)32.7(sincos

)()31.7(cos)(

)30.7(cossin

)()29.7(sin)(

10

10

zz

zzzz

zz

zzzz

+=−−−=

−=−−−=

χχ

ψψ

7.8. EVALUATION OF RICATTI – BESSEL FUNCTION )(αψ n

Putting n=1 in the recurrence formula of cylindrical function, one gets

)(3

)()( 120 zz

zz ψψψ =+ ⇒ zz

zz

z cos)3

(sin)13

()(22 −+−=ψ (7.33)

Similarly n=2 in the recurrence formula of cylindrical function gives rise to

)(5

)()( 231 zz

zz ψψψ =+ ⇒ zz

zzz

z cos)15

1(sin)615

()(233 −+−=ψ (7.34)

The higher orders of the function can be similarly worked out as follows:

zzz

zzz

z cos)10105

(sin)145105

()(3244 +

−++−=ψ (7.35)

zzz

zzzz

z cos)1945105

(sin)15420945

()(42355 −−++−=ψ (7.36)

zzzz

zzzz

z cos)21103951260

(sin)1210472510395

()(532466 −−+−+−=ψ (7.37)

Thus the Ricatti-Bessel function )(znψ can be expressed as equation (7.38) in terms of

the trigonometric functions sin and cos with polynomial functions of the size parameter As

and Bs as their variable coefficients.

114

znBsznAszn cos)(sin)()( +=ψ (7.38)

The number of terms needed for the evaluation of scattering efficiency can be

determined only after seeing the convergence. It is observed that six to seven terms of the

Ricatti Bessel functions gives sufficient accuracy. Hence the details of derivations of seven

terms of these functions are explained below for provide the computational details of the

scattering efficiency factor.

Since the suffix zero cannot be put as suffix in a FORTRAN program, the order of

Ricatti-Bessel function is incremented by an order 1 in representation. ie; )(znψ where

n=0 to 6 defined in the mathematical form are renamed from n=1 to 7 respectively in the

numerical algorithm as shown in the table 7.1. This is done for the ease of computer

programming.

Table 7.2 Coefficients As(n)& Bs(n) of )(znψ of eqn. 7.38

Term, n Coefficient of sin function, As(n) Coefficient of cosin function, Bs(n)

1 1.0 0.0

2

z

1

-1.0

3 1

32

−z

z

3−

4

zz

6153

− 2

151

z−

5 1

4510524

+−zz

zz

101053

+−

6 +−

35

420945

zz z

15 1

94510542

−−zz

7 1

210472510395246

−+−zzz

zzz

2110395126053

−−

115

7.9. EVALUATION OF RICATTI FUNCTION )(znχ

The following initial values are used to generate the higher orders as explained below:

)40.7(sincos

)(

)39.7(cos)(

1

0

−−−+=

−−−=

zz

zz

andzz

χ

χ

Putting n=1 in the recurrence formula of cylindrical function, one gets

)(3

)()( 120 zz

zz χχχ =+ ⇒ zz

zz

z cos)13

(sin)3

()(22 −+=χ (7.41)

Similarly n=2 in the recurrence formula of cylindrical function gives rise to

)(5

)()( 231 zz

zz χχχ =+ ⇒ zzz

zz

z cos)615

(sin)115

()(323 −+−=χ (7.42)

The higher orders of the function can be similarly worked out as follows:

zzz

zzz

z cos)145105

(sin)10105

()(2434 +−+−=χ (7.43)

zzzz

zzz

z cos)15420945

(sin)1105945

()(35245 +−++−=χ (7.44)

zzzz

zzzz

z cos)1210472510395

(sin)21103951260

()(246536 −+−+++−=χ (7.45)

Thus as in the case of the Ricatti-Bessel function )(znψ , )(znχ also can be expressed

as equation (7.46) in terms of the trigonometric functions sin and cos with polynomial

functions of the size parameter As and Bs as their variable coefficients.

znBkznAkzn sin)(cos)()( +=χ (7.46)

As in the case of first Ricatti-Bessel function, here also, the above order of terms of second

Ricatti Bessel function are incremented by an order 1 for the ease of programming. ie;

)(znχ where the suffix n=0 to 6 are renamed from n=1 to 7 respectively as shown in the

table 7.3.

116

Table 7.3 COEFFICIENTS Ak(n) & Bk(n) of )(znψ of eqn. 7.46

Term, n Coefficient Ak(n) Coefficient Bk(n)

1 1.0 0.0

2

z

1

1

3 1

32

−z

z

3

4

zz

6153

− 115

2−

z

5 1

4510524

+−zz

zz

101053

−

6 +−

35

420945

zz z

15 1

10594524

+−zz

7 1

210472510395246

−+−zzz

zzz

2112601039535

+−

It may be noted from Table7.2 and Table7.3, that both the Ricatti Bessel functions

expressed in the form of equations (7.38) and (7.46) can be expressed by the same

coefficients as shown below:

znBsznAszn cos)(sin)()( +=ψ (7.38)

znBkznAkzn sin)(cos)()( +=χ (7.46)

Where

)48.7()()(

)47.7()()(

nBsnBk

nAsnAk

−=

=

This similarity of expressions obtained for both the Ricatti-Bessel functions considerably

simplifies the coding of the computer program for predicting scattering efficiency factors.

117

7.10. DERIVATIVES OF RICATTI BESSEL FUNCTIONS

7.10.1. DERIVATIVE of )(znψ

The function ( ) ( ) sin ( ) cosn

z As n z Bs n zψ = +

is differentiated with respect to the argument z to obtain the derivatives )(zn

′ψ . Since

As(n) and Bs(n) are functions of the size parameter z ,obviously,

; ( ) ( ( ))sin ( ) cos ( ( ))cos ( ))sinn

d die z As n z As n z Bs n z Bs n z

dz dzψ ′ = + + − (7.49)

[ ] [ ]

)52.7()()()(

)51.7()()()(

)50.7(cos)(sin)(

cos)()(sin)()(

cos)())((sin)())(()(;

nAsnsBnBd

andnBsnsAnAdwhere

znBdznAd

znAsnsBznBsnsA

znAsnBsdz

dznBsnAs

dz

dzie n

+′=

−′=

+=

+′+−′=

++

−=

′ψ

Equations (7.51) and (7.52) are evaluated for n=1,2,…7 and the values of Ad(n) and Bd(n)

are listed in table 7.4 .

Table 7.4 COEFFICIENTS Ad(n) & Bd(n) of )(zn

′ψ of eqn 7.50

Term, n Coefficient of sin function, Ad(n) Coefficient of cosin function, Bd(n)

1 0 1.0

2 1

12

+−

z

z

1

3

ZZ

363

+−

16

2−

Z

4 1

452142

−−zz

zZ

6453

−

5

Zzz

1019542035

−+−

155420

24+−

zz

6 1

12022054725246

+−+−

ZZZ

ZZZ

15630472535

+−

7

ZZZZ

2116802929562370357

+−+−

1231850562370

246−+−

ZZZ

118

7.10.2. DERIVATIVE OF )(znχ

Similarly, the function znBkznAkzn sin)(cos)()( +=χ is

differentiated with respect to z as follows:

[ ] [ ]

)56.7()()()(

)55.7()()()(

)54.7(cos)(sin)(

sin)()(cos)()(

)53.7(cos)(sin))(()sin)((cos))(()()(

nBknkAnBkd

andnAknkBnAkdwhere

znBkdznAkd

znAknkBznBknkA

znBkznBkdz

dznAkznAk

dz

d

dz

dz nn

+′=

−′=

+=

−′++′=

++−+==′ χχ

Equations (7.55) and (7.56) are evaluated for n=1,2,…7 and the values of Akd(n) and

Bkd(n) are listed in table 7.5 .

Table 7.5 COEFFICIENTS Akd(n) & Bkd(n) of )(zn

′ψ of eqn 7.54

Term, n Coefficient Akd(n) Coefficient Bkd(n)

1 -1 0

2 -

z

1 1

12

+−

z

3 )1

6(

2−−

Z

ZZ

363

+−

4 )

645(

3zZ

−− 14521

42−−

zz

5 )1

55420(

24+−−

zz

Zzz

1019542035

−+−

6 (

+−−

ZZZ

15630472535

) 112022054725

246+−+

−

ZZZ

7 -( 1

231850562370246

−+−ZZZ

) ZZZZ

2116802929562370357

+−+−

It may be noted from Table7.4 and Table7.5, that derivatives of both the Ricatti Bessel

functions expressed in the form of equations (7.50) and (7.54) can be expressed by the

same coefficients as shown below:

119

)57.7(cos)(sin)()( znBdznAdzn +=′ψ

)58.7(cos)(sin)()( znBkdznAkdzn +=′χ

Where )60.7()()(

)59.7()()(

nBkdnBd

nAkdndA

−=

=

The similarity of expressions of terms of both the Ricatti Bessel functions is also

seen in the case of their derivatives as shown by equations (7.59) and (7.60). This property

of terms of the derivatives of Ricatti Bessel functions also considerably simplifies the coding

of the computer program for predicting scattering efficiency factors.

7.11. EVALUATION OF SCATTERING EFFICIENCY

The trigonometric functions tan(αn) and tan(βn) can be evaluated by solving the

equations (7.38) and (7.46) for )(znψ and )(znχ respectively and equations (7.50) and

(7.54) for their derivatives. Once tan(αn) and tan(βn) are known, sin2(αn) and sin2(βn) can be

directly calculated to obtain the scattering efficiency given by the equation (7.19).

7.12. SCATTERING IN A CONTROL VOLUME CONTAINING MANY PARTILCES

According to Lamberts law, the monochromatic radiant energy traverses a random,

attenuating medium where the probability of a photon experiencing an absorption/scattering

event, the radiant power flux decays exponentially with the product of the path length in the

medium and the attenuation coefficient of the medium. Lamberts law is a limiting case of the

steady state RTE for a medium with no internal sources in which the path function of the

RTE vanishes. In the present context, the path function is the radiance gained per unit

distance, s, by scattering of the radiance field at s in the direction of the axis. Hence

attenuation due to scattering of solid particles scattered in random fashion in each control

volume of the computational domain should be defined.

When the attenuation coefficient is expressed in terms of the concentration of

particles, the formula for the exponential decay of radiation power is known as the Beer

Law. The scattering efficiency of a volume cell with a number of different sized particles is

120

depending on the concentration of particles. It has been found (Van de Hulst PP: 32) that,

for quite different reasons, the scattering and extinction cross sections of the single particles

must be added to give the corresponding cross sections for the entire cloud. It also states

that the scattering pattern caused by reflection on very large convex particles with random

orientation is identical with the scattering pattern by reflection on a very large sphere of the

same material and surface condition (page:111, Vande Hulst)

Let there be )(rdN particles per unit volume in the radii ranging from r to r+dr and

let Csca be the scattering cross section for a particle of radius r. Then integrating over all the

particles, the effective scattering area in a differential volume dV of the particle cloud can be

found as

)()(0)(

rdNrCdVCrN

scasca ∫∞

==

Equivalently, for particles of sizes of m different categories, the scattering efficiency of the

cell is defined as

( )2

1

isi

m

i

c rNcQQ π∑=

=

where Qsi is the scattering efficiency of the ith category of particles

Nc is the number of particles of the ith categrory present in the cell

ri is the radius of the ith category of particles

Here the number of particles, Nc, in each cell is obtained from the particle density

derived from the particle spectrum as per the Rosin-Rammler distribution.

7.13. EXPERIMENTAL VALUES OF EXTINCTION COEFFICIENTS

Experimental Values of Extinction Coefficients, T.E. Mills, PJ Bishop and A.Minardi

[1994] of Laser-Produced Aluminium Plumes were obtained as a function of radial position

and time at an axial location. The purpose of the study was to measure the extinction

coefficient of a two dimensional, axisymmetric laser produced aluminum cylindrical plume at

one wavelength 0.632 µm. The choice of wavelength was made due to the cost efficiency

and ease of operation provided by He:Ne lasers. A He:Ne probe laser scanning revealed

that the aluminium plume is optically thin and that the radial extinction coefficient profile has

a Gaussian distribution. Extinction coefficients were measured as a function of radial

position and time at one axial location. It was found that the extinction coefficients

121

decreased with radial distance from the laser beam center. The maximum extinction

coefficient measured at the center of the laser beam is 0.6 cm-1 and this corresponds to an

optical thickness of about 0.04. However, the authors give precautions that this technique

for measurement of extinction coefficients cannot be used when liquid droplets are present

in high concentrations.

7.14. COMPUTED SCATTERING EFFICIENCY PARAMETERS

Wavelength of the incident radiation is estimated from the universal form of spectral

distribution of blackbody hemispherical emissive power as discussed in chapter 4.

Refractive index of the alumina particle can be taken from the curve shown in fig.7.16. At a

particular axial location of the domain of computation, variation of size parameter is

observed as ranging from 0.46 to 8.76. Fig.7.17 shows typical values of the scattering

efficiency factors evaluated in the computational domain of the present study.

Even though the observations made in this experiment might not be valid for large

solid rocket motors, these observations could be checked in this theoretical study. Figure

7.8 shows the predicted scattering efficiency factor of control volumes defined in the radial

direction from the centre of plume at a typical axial location. This graph is corresponding to

a case where ten control volumes are defined radially. More continuity will be obtained if

more number of control volumes is defined in the radial direction. Even though all types of

particles are contained in the first few control volumes, the scattering efficiency of these

cells are lower because of their lower volumes and thereby lower number of particles.

Similarly in the case of control volumes defined in the outer region and its periphery, the

particle size has come down causing lower scattering efficiency. Further it may be noted

that, the trend of the curve may be approximated to that of a Gaussian distribution starting

from the centre of the plume.

122

0

2.5

5.0

7.5

10.0

12.5

1.2 1.3 1.4 1.5 1.6

Radial distance(m)

Ne

t s

catt

eri

ng

ex

tin

cti

on

fa

cto

r (m

-1)

Figure7.8 Radial scattering efficiency factor from the centre of the plume

7.15. SCATTERING CHARACTERISTICS OF EXHAUST PLUME

The scattering characteristics of exhaust plume and the aluminium oxide particles

are to be studied for estimating the scattering efficiency of solid particles contained in the

plume. They are

(1) Size and shape of aluminum oxide particles contained in the plume

(2) Refractive indices of different constituents of the exhaust plume

(3) Refractive index of the aluminium oxide particles.

Dewban, Kinslow and Watson [1988] reveals that the dust particles from 0.01 to 0.1

µ are irregularly shaped and in many cases have agglomerated into clusters. Those

particles in the range from 1 to 100 µ are spherical. It is quite common to find some of the

smaller spheres attached to the larger ones. Thus the geometrical shapes of aluminium

particles in the exhaust plume of rockets can be very well approximated to spheres of

different radii. Thus the formulation for scattering efficiency of spherical particles can be

adopted in the study.

Refractive indices of different gaseous components in the exhaust plume are listed

in Table 7.6. This table indicates that the refractive index of the gaseous constituents can be

approximated as 1. A discussion on refractive index of alumina is given by Edwards and

123

Babikian [1990] and the finally converged values are published by the website WIKIPEDIA

due to Boston [1991] and is shown in figure 7.9. Thus the ratio of refractive index of

Alumina particles to that of gaseous constituents of the exhaust plume becomes numerically

equal to the refractive index of alumina. Study on particle spectrum of alumina shows that

percentage of particles greater than 5µm is negligible and hence the diameter of particles

taken in this study is in the range 0.5 to 5µm. Wave length of the incident energy is taken

from the spectrum of radiant energy discussed in chapter 4. Thus particle size parameter

and refractive index are defined for predicting the scattering efficiency of the alumina

particles in the exhaust plume of solid rockets.

Table 7.6 Refractive indices of combustion gases

Species Refractive

index

Water vapor 1.000256

Carbon dioxide 1.000449

Carbon monoxide 1.000338

Oxygen 1.000271

Hydrogen 1.000132

Nitrogen 1.000298

Chlorine 1.000773

Air 1.000292

Nitric Oxide 1.000297

Hydrochloric acid 1.000447

124

1.725

1.750

1.775

1.800

1.825

1.850

0 0.5 1.0 1.5 2.0 2.5

Al2o

3 solid particles

Wavelength(µµµµm)

Re

fra

cti

ve

in

de

x

Fig. 7.9 Spectral variation of refractive index of Alumina particles

7.16. RESULTS AND DISCUSSIONS

Since it is often required to mention the developed mathematical model of Mie

scattering, it will be referred as Mie model in this section. It is essential to validate the

developed Mie model. This is accomplished by comparing the predicted values of scattering

efficiency by the Mie model with standard literature. For non-absorbing spheres, Van de

Hulst has derived a formula to define the salient features of the extinction curve not only for

m close to unity but even for values of m as large as 2. The formula for scattering efficiency

factor for a spherical particle is

12)cos1(4

sin4

22

−=−+−= mzwhereQext ρρρ

ρρ

, (7.61)

z is the size parameter and m is the ratio of the refractive index of the scatterer with that of

the medium in which the scatterer is located. Scattering does not occur for m=1. The

developed Mie model encounters a singularity at m=1 and hence no value is delivered by it

for m=1. For m nearer to 1, scattering is small and this is predicted by the model as can be

seen in fig 7.10.

125

0

0.1

0.2

0.3

0 2 4 6 8

Aproximate formulaMie Theory

Size Parameter

Sca

tte

rin

g E

fic

ien

cy F

acto

r

Fig.7.10: Comparison between Mie model and approximate formula for m=1.05

Even for a higher value of the size parameter of 8, the scattering efficiency factor is

only 0.3. Again it may be seen that both approximate formula and the Mie model almost

coincides at the majority of the size parameters except at the beginning and ending of the

domain of the function. Figure 7.11 is the comparison of scattering efficiency factors

predicted by Mie model and the approximate formula. It may be noted that when the value

of m is increased from 1.05 to 1.1, the scattering efficiency factors have increased almost

three times. The slight differences between the values of efficiency factors predicted by Mie

model and approximate formula starts from the size parameter value of 5.5

126

0

0.25

0.50

0.75

1.00

1.25

0 2 4 6 8

Mie theoryApproximate formula

Size parameter

Sc

att

eri

ng

eff

icie

nc

y f

ac

tor

Fig.7.11 Comparison between Mie model and approximate formula for m=1.1

Figure 7.12 shows the difference in the trends of increase of scattering efficiency factors for

the values of m=1.05 and 1.1. The equation of curve for the efficiency factors as a function

of the size parameter for the case m=1.05 is

00444.000377.000412.0)( 2 ++= zzzQext (7.62)

Whereas, the equation of curve for the efficiency factors as a function of the size parameter

for the case m=1.10 is

00830.00405.00124.0)( 2 −+= zzzQext (7.63)

Equations (7.62) and (7.63) show that the scattering efficiency factors are almost linearly

increasing with the increase of size parameters, while for m=1.1, the increase is of non-

linear nature.

According to Van de Hulst, If the absolute value of (m-1) is very small, ie;

,1 smallveryiswherem δδ≈− it makes no difference in the scattering pattern

whether 0101 ≺ −− morm . Figures 7.13 and 7.14 are provided to study the

127

range of the domain of size parameter for validity of this theory. It can be seen from these

figures that up to the value of z=4.5, the values of scattering

0

0.4

0.8

1.2

0 2 4 6 8

Mie theory (m=1.1)Aproximate formula (m=1.1)Mie theory (m=1.05)Approximate formula (m=1.05)

Size parameter

Sc

att

eri

ng

eff

icie

nc

y f

ac

tor

Figure 7.12 Comparison of scattering efficiency factors for the values of

m=1.05 and 1.1.

0

0.1

0.2

0.3

0 2 4 6 8

m=0.95m=1.05

Size parameter

Sc

att

eri

ng

eff

icie

nc

y f

ac

tor

Figure 7.13 Comparison of scattering efficiency factors for the values of m=0.95 and

m=1.05

128

Efficiencies are matching for the case 05.01 =−m and then onwards start deviating.

Again the values of scattering matches up to the value of z=3.0 and then starts deviating for

the case 10.01 =−m . This indicates that as the value of refractive index starts deviating

more from the value 1, the domain of size parameter where the same scattering pattern

exists is reduced. This validates the view of Van de Hulst.

0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8

m=1.1m=0.9

Size parameter

Scatt

eri

ng

eff

icie

ncy

Figure 7.14 Comparison of scattering efficiency factors for the values of

m=1.1 and m=0.9

Figures 7.15 and 7.16 give the predicted scattering efficiency factors by Mie theory

for m=1.5 and m=2.0 respectively and their comparison with the approximate formula. It

may be seen from figures 7.15 and 7.16 that up to the value of particle size of 0.7, the trend

of Mie theory values are just opposite to that of approximate formula. Then afterwards, both

are in the same trend and Mie theory captures more wiggles in the values of scattering

efficiency. Figure 7.17 gives the comparison of scattering efficiency factors with the

approximate formula.

129

0

2

4

6

0 2 4 6 8

Approximate formulaMie theory

Particle size

Sca

tteri

ng

Eff

icie

nc

y f

ac

tor

Fig. 7.15 Comparison of values of scattering efficiency factor for m=1.5

0

2

4

6

8

10

0 2 4 6 8

Mie theory (m=2.0)Approximate formula (m=2.0)

Particle size

Sca

tteri

ng

eff

icie

ncy

fac

tor

Fig 7.16 Comparison of values of scattering efficiency factor for m=2.0

130

0

1

2

3

4

0 2 4 6 8

Approximate formula (m=3.0)Approximate formula (m=2.0) Approximate formula (m=1.5)

Particle size

sc

att

eri

ng

eff

icie

nc

y f

ac

tor

Fig. 7.17 scattering efficiency factor predicted by approximate formula

0

4

8

12

16

0 2 4 6 8

Mie theory (m=3.0)Mie theory (m=2.0)Mie theory (m=1.5)

Particle size

Sc

att

eri

ng

eff

icie

nc

y f

ac

tor

Fig. 7.18 scattering efficiency factor predicted by Mie theory

7.16.1. Sensitivity of Scattering Efficiency on Wavelength

Figures 7.19 and 7.20 shows the variation of scattering efficiency of particles of 5

µm and 2 µm radii with a refractive index of 1.80. Both curves show differences in scattering

efficiency for lower values of wavelength, but show the same trend of approximate theory.

131

Mie theory could capture more number of wiggles as seen in these curves. For the higher

size particle, scattering efficiency exhibits an increasing trend with the increase of

wavelength as seen in figure 7.19. Whereas, for the lower size particle, scattering efficiency

exhibits an increasing followed by a decreasing trend with the increase of wavelength as

seen in figure 7.20

0

2

4

6

8

10

0.2x10-5

0.4x10-5

0.6x10-5

0.8x10-5

1.0x10-5

0

2

4

6

0

3

6

9

12

15

size parameterMie theory (5µ)Approximate formula

Wavelength,m

Sc

att

eri

ng

eff

icie

nc

y

Fig.7.19 Sensitivity of scattering efficiency of a particle of radius 5 µ on wavelength

0

2

4

6

8

10

0.35x10-5

0.70x10-5

1.05x10-5

0

5

10

15

Mie theory (radius=2µ)Approximate formulaSize parameter

Wavelength (m.)

Sc

att

eri

ng

eff

icie

nc

y

Fig.7.20 Sensitivity of scattering efficiency of a particle of radius 2 µ on wavelength

7.17. CLOSURE

The required fundamental parameters for modeling the scattering of radiant energy

are described in this chapter. Scattering laws apply with equal validity to all wavelengths.

Interestingly, these depend upon the size parameter which is the ratio of the circumference

132

of particle to the wavelength of incident radiation. Depending on the range of values of this

natural size parameter, different theories exist to define the scattering efficiencies. Both

Rayleigh and Mie theory of scattering are described in this context. The scattering cross

sections and efficiencies of spherical particles of Al2O3 are estimated as a function of their

size parameter and their dependence on wavelength of the incident energy is separately

addressed. Rayleigh formula is a single formula showing its inverse dependence to the

fourth power of the wavelength. The Mie scattering is valid over a wide range of the size

parameter. Theory of Mie scattering makes use of Ricatti-Bessel functions because of their

asymptotic behavior for capturing the wiggles in the curves of scattering efficiencies.

Approximate formula exist for scattering efficiencies in certain range of the size parameter

and is used for comparing the scattering efficiencies predicted by Mie theory in the limited

range where the formula is valid . It is observed that the trends of the curves generated by

the two methods are the same and show reasonably good comparison in certain region.

However the approximate formulae are not able to capture the wiggles for higher particle

sizes. The minor wiggles appearing in the two curves infer that scattering efficiency vary

drastically and highly non-linear with the wavelength and size parameter so that prediction

of scattering efficiency by interpolating even in the close intervals becomes erroneous.

In general, the scattering efficiency is used as an input for various analysis and

hence simple and fast computing model is preferred. For this, the Ricatti Bessel functions

and their derivatives appearing in the Mie coefficients are expressed in terms of polynomials

of the size parameter. These functions are derived up to the 7th term using their recurrence

relations to obtain good convergence. In general, the scattering efficiency is used as an

input for various analysis and hence simple model is preferred. Ideally, the subroutines

made from these models should take less computer time without reducing the accuracy.

This objective could be met in this study by expressing the Mie coefficients with the

polynomials of size parameter. Major findings of this chapter is published as a paper titled

“Modelling of Scattering in Thermal Radiation from Aluminium Oxide Particles in the

Exhaust Plume of a Solid Motor ” in the proceedings of 19th National and 8th ISHMT-ASME

Heat and Mass Transfer Conference , 03-05,Jan,2008, JNTU, Hyderabad.